GNR607 Principles of Satellite Image Processing Instructor: Prof. - PowerPoint PPT Presentation

GNR607 Principles of Satellite Image Processing Instructor: Prof. B. Krishna Mohan CSRE, IIT Bombay bkmohan@csre.iitb.ac.in Slot 2 Lecture 06 Mathematical Preliminaries - 1 Aug. 04, 2014 9.30 AM – 10.25 AM

IIT Bombay Slide 1 Aug 04, 2014 Lecture 06 Math. Preliminaries - 1 Contents of the Lecture • Mathematical Preliminaries – Matrix Operations – Vectors – Eigenanalysis of matrices GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 2 Review of Mathematical Preliminaries • All remote sensing related data processing and analysis require knowledge of mathematics • ALL areas of mathematics are relevant in processing images acquired by remote sensing • Topics considered for a brief review: – Matrix vector operations and eigenvalue problem – Probability and Statistics – Linear System Principles GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 3 Mathematical Preliminaries GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 4 Language of Image Processing The language of digital image processing is mathematics – Linear Algebra – Optimization – Probability Theory and Statistics – Matrices and vectors – Geometry – Integral and differential equations – Fuzzy Sets GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 5 Matrices and vectors • Any matrix A of size M x N is an array of entities or elements (symbols, real numbers, integers, complex numbers …) having M rows and N columns   a a ... a 11 12 1 N   10 20 30 40 20 30 Red: Elements on   15 10 20 15 15 10 main diagonal a a ... a 30 20 40 20 11 19   21 22 2 N =  A 33 40 51 98 10 12  4x4 Square 4x2 Rectangular ...   Trace = 10+10+40+98=158   a a ... a M 1 M 2 MN     GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 6 Definitions • Matrix A is rectangular if M ≠ N; else it is a square matrix • An element a mn is said to be on the main diagonal of a square matrix A if m = n, i.e., its row and column indices are the same • If all elements a mn of a square matrix A are zero for m ≠ n, then A is called a diagonal matrix 10 0 0 0 22 0 • A is a null matrix if a mn = 0 for 1  m  M, 1  n  N 0 0 44 M ∑ a Diagonal • Trace (A) = where A is a square matrix of size M mm matrix = m 1 0 0 0 • Matrix A = Matrix B iff a mn = b mn , 1  m  M, 1  n  N 0 0 0 Null GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 7 Definitions • The transpose of a matrix A, denoted by A T is produced by interchanging the rows and columns of A. If A has M rows and N columns, A T will have N rows and M columns Symmetric Matrix 15 41 30 15 22 12 30 41 22 39 50 41 39 30 23 50 30 50 41 50 06 Matrix Transpose • If A = A T , then A is called a symmetric matrix • If we multiply every element a mn of matrix A by a scalar k, then k. A is called the scalar multiple of A where k may be real or complex. If k = -1, then the resultant is known as the negative of A 15 22 31 30 44 62 -15 -22 -31 44 10 27 88 20 54 -44 -10 -27 Matrix A 2A -A GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 8 Definitions • A matrix A of size M x N is called a column vector if the number of columns N = 1 • A matrix B of size M x N is called a row vector if the number of rows M = 1 Column vector   a is a matrix with 1   only one a   [ ] [ ] T = 2 = = column, row a b b b ... b a a a ... a   1 2 N 1 2 M ... vector is a   matrix with a   only one row M GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 9 Basic Operations • Matrix A obtained by the sum of two matrices B and C is defined as an array of elements a mn a mn = b mn + c mn • Matrix summation is possible only if both matrices B and C have the same number of rows and columns M and N • If A is the difference of matrices B and C, then a mn = b mn - c mn GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 10 Basic Operations • Matrix A obtained by the product of matrices B MxN and C NxP is of size MxP, whose (i,j) th element a ij is defined by N If B.C is feasible to calculate, C.B ∑ b c can be calculated only if B and C • a ij = in nj are square matrices = n 1 • Matrix multiplication is only possible if the number of columns of the first matrix = the number of rows of the second matrix. 12 15 22 09 10 30 0 462 460 1009 59 16 20 14 x 06 08 11 1 = … 09 12 10 12 10 22 2 13 15 17 4x3 matrix 3x4 matrix 4x4 matrix GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 10a Basic Operations • The inverse of a square matrix A, denoted by X satisfies the property A.X = X.A = I where I is the unit matrix If the inverse does not exist for a matrix A, then it is non-invertible. In such a case A is called a singular matrix. 12 15 22 09 10 30 0 462 460 1009 59 16 20 14 x 06 08 11 1 = … 09 12 10 12 10 22 2 13 15 17 4x3 matrix 3x4 matrix 4x4 matrix GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 10b Basic Operations • If A is a rectangular matrix, a conventional inverse of A does not exist. Instead, one computes a pseudo-inverse of A. In case of pseudo-inverse, there will be two such matrices for a rectangular matrix A. If A + denotes pseudo-inverse of rectangular matrix A of size MxN, then there will be a left pseudo-inverse resulting in ( A + ) L .A = I , of size NxN, and a right pseudo-inverse resulting in A.(A + ) R = I of size MxM GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 10c Basic Operations Pseudo-Inverse Usually such situations are encountered when it is required to solve systems of equations with a)Number of equations more than number of unknowns e.g., A.p = q where A is of size 10x3, p is the unknown vector of size 3x1, and q is of size 10x1. Number of equations = 10 and number of unknowns = 3 b) Number of equations less than number of unknowns e.g., B.r = s where B is of size 3x5, r is the unknown vector of size 5x1, and s is of size 3x1. Number of equations = 3 and number of unknowns = 5 GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 11 Vectors and Vector Spaces • A vector space is defined as a non-empty set of vectors V that satisfy properties as shown below: • Vector addition – x + y = y + x – x + (y + z) = (x + y) + z – 0 + x = x + 0 = x (0 is known as zero-vector ) – -x + x = x + (-x) = 0 (-x is called negative of x) GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 12 Vectors and Vector Spaces Multiplication condition • p ( q y) = ( pq )y given scalars p and q , and vector y. • p (x + y) = p x + p y given scalar p and vectors x and y. • ( p + q ) x = p x + q x given scalars p and q , and vector x. Multiplicative identity • 1 x = x 1 = x for all vectors x --- multiplying a vector by unity does not alter it. GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 13 Linear Combination of Vectors • Given two vector spaces V 1 and V 2 , if all elements of V 2 are also elements of V 1 , then V 2 is said to be a subspace of V 1 • Linear combination of vectors v k , k = 1,2, … is defined as v = w 1 v 1 + w 2 v 2 + … + w n v n The weights w 1 , w 2 , … are scalars GNR607 Lecture 06 B. Krishna Mohan

IIT Bombay Slide 14 Inner Product • The inner product of two column vectors a and b of size n is a scalar, given by • a.b = a T b = b T a = a 1 b 1 + a 2 b 2 + … + a n b n n ∑ a b i i • This can also be written as a.b = = i 1 • Inner product is also referred to as vector dot product . GNR607 Lecture 06 B. Krishna Mohan